Monday, March 31, 2008

proofs as explanations

Most proofs that I've seen feel like defensive, motley soups of disparate thoughts whose composition does indeed imply the theorem in question; but the reader is left without a strong intuition as to why the fact is true.

To illustrate: Consider one of the most fundamental theorems of mathematics: the Pythagorean. Given a right triangle with sides of lengths a, b, and hypotenuse c, we must have a2 + b2 = c2. Euclid, in The Elements, presented an argument for a proof involving many steps (14, by wikipedia's current count) which involves showing the similarity of several triangles and the equivalence in area of various different shapes. Check out wikipedia for the full proof.

Euclid's proof is mathematically sound. It achieves its aim of demonstrating, beyond a doubt, that in fact a2 + b2 = c2 for all (Euclidean) right triangles. What it leaves wanting is intuition. Why? How can a human, without resorting to memorization, understand that the sides must relate in this manner, upon seeing any right triangle? [The word grok comes to mind.]

I think that some proofs do achieve this higher standard of explaining the fact that they defend. Here, in a single picture, is a complete proof of the Pythagorean:

If I were a teacher, trying to convey as best I could the intuition behind this theorem, I would animate this figure so that the ratio a/b transitioned from 0 to infinity and back repeatedly -- the point being to further the deep comprehension that the equality in area between the squares in question is independent of the acute angles of the triangle.

To give another example: one way to view De Moivre's theorem is

cos (nx) = Real[ (cos x + i sin x)n ].

De Moivre proved this before Euler's formula, eix = cos x + i sin x, was known. One proof follows by induction on n -- assume it holds for n, then show it still holds for n+1. (The base case, n=1, is trivial.)

Suppose you saw this formula and the proof by induction. Would you really say that you knew why the formula was true? I am guessing that most people, including mathematicians, would gain profound new insight into this theorem when seen in terms of Euler's formula. In that case, we should restate De Moivre's theorem as

(cos x + i sin x)n = cos(nx) + i sin(nx),

and it becomes a very straightforward corollary of Euler's formula:

(eix)n = ei(nx).

With Euler's formula, De Moivre is just one perspective of a special case; it's a shadow on our allegorical wall, while Euler shows us the true form of the fact.

Most math work today is presented as if intuition were a burden to be borne rather than the enabler of our creativity. Gauss in particular is infamous for hiding the how-d'you-think-of-that of his work. Perhaps the math community as a whole would move along more constructively if we embraced the major role intuition and deep understanding play in our creative efforts.

One does not discover a complex path of reasoning without a guess at the lay of the land. Why ignore the curve of the lines while we strive to connect the dots?